(Solved): d. Define the chi-square distribution \chi _(n)^(2) with n degrees of freedom as the sum of n inde ...
d. Define the chi-square distribution
\chi _(n)^(2)with
ndegrees of freedom as the sum of
nindependent standard normal random variables. That is,
\chi _(n)^(2)is distributed as
\sum_(i=1)^n Z_(i)^(2)where
Z_(i)?N(0,1)iid. Let
x_(i)be
nindependent normal random variables
N(\mu ,\sigma ^(2))iid. In this section, prove that
V?\sum_(i=1)^n (x_(i)-(\bar{x} ))^(2)(1)/(\sigma ^(2))?\chi _(n-1)^(2)(i.e.,
Vis chi-square distributed with
n-1degrees of freedom). Define the random vector
x=[x_(1),dots,x_(n)]^(T)inR^(n)and explain why it is multivariate normal. (2 points). There exists an orthogonal matrix
BinR^(n\times n)such that its first row consists of the values
(1)/(\sqrt(n))(such a matrix exists according to the Gram-Schmidt process). Define the random vector
Y=
(1)/(\sigma )B(x-\mu 1)where
1=[1,1,dots,1]^(T)inR^(n). Show that
Y?N(0,I)and compute the first component
Y_(1)as a function of
\bar{x} . (2 points). Compute
Y^(T)Y(using the fact that
Bis orthogonal) and prove that
Vis chi-square distributed with n-1 degrees of freedom. (Use the expression for
Y_(1)from section 2.) (2 points). please don't provide chatGPT answers and try to explain each step the way they want the proof.
